4 Theorem 1.5.1 Row Operations by Matrix Multiplication If the elementary matrix E results from performing a certain row operation on and if A is an m×n matrix ,then the product EA is the matrix that results when this same row operation is performed on A .When a matrix A is multiplied on the left by an elementary matrix E ,the effect is to performan elementary row operation on A .
6 Inverse OperationsIf an elementary row operation is applied to an identity matrix I to produce an elementary matrix E ,then there is a second row operation that, when applied to E, produces I back again.Table 1.The operations on the right side of this table are called the inverse operations of the corresponding operations on the left.
7 Example3 Row Operations and Inverse Row Operation The 2 ×2 identity matrix to obtain an elementary matrix E ,then E is restored to the identity matrix by applying the inverse row operation.
8 Theorem 1.5.2Every elementary matrix is invertible ,and the inverse is also an elementary matrix.
9 Theorem 1.5.3 Equivalent Statements If A is an n×n matrix ,then the following statements are equivalent ,that is ,all true or all false.
10 Row EquivalenceMatrices that can be obtained from one another by a finite sequence of elementary row operations are said to be row equivalent .With this terminology it follows from parts (a )and (c ) of Theorem that an n×n matrix A is invertible if and only if it is row equivalent to the n×n identity matrix.
12 Example4 Using Row Operations to Find (1/3) Find the inverse ofSolution:To accomplish this we shall adjoin the identitymatrix to the right side of A ,thereby producing amatrix of the formwe shall apply row operations to this matrix untilthe left side is reduced to I ;these operations willconvert the right side to ,so that the final matrixwill have the form